The Space of Closed $G_2$-Structures. I. Connections

Abstract

In this article, we develop foundational theory for geometries of the space of closed $G_2$-structures in a given cohomology class as an infinite-dimensional manifold. We introduce Sobolev-type metrics, construct their Levi-Civita connections, formulate geodesic equations, and analyse the variational structures of torsion free $G_2$-structures under these Sobolev-type metrics.

51 pages

References in corpus (3)

• Laplacian Flow for Closed $G_2$-Structures: Short Time Behavior
• Hodge Theory for G2-manifolds: Intermediate Jacobians and Abel-Jacobi maps
• Local geometry of the G2 moduli space